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Investigation of Thermal Adiabatic Boundary Condition on Semitransparent Wall in Combined Radiation and Natural Convection

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Investigation of Thermal Adiabatic Boundary Condition on Semitransparent Wall in Combined Radiation and Natural Convection Investigation of Thermal Adiabatic Boundary Condition on Semitransparent Wall in Combined Radiation and Natural Convection G Chanakya, Pradeep Kumar Numerical Experiment Laboratory (Radiation and Fluid Flow Physics) School of Engineering Indian Institute of Technology Mandi Mandi, Himachal Pradesh, India 175075 Abstract Two thermal adiabatic boundary conditions arise on the semitransparent win- dow owing to the fact that whether semitransparent window allows the energy to leave the system by radiation mode of heat transfer. It is assumed that being low conductivity of semitransparent material, energy does not leave by conduction mode of heat transfer. This does mean that the semitransparent window may behave as only conductively adiabatic (qc = 0) or combinedly con- ductively and radiatively adiabatic (qc +qr = 0). In the present work, the above two thermal adiabatic boundary conditions have been investigated in natural convection problem for the Rayleigh number (Ra) 105 and Prandtl number(Pr) 0.71 in a cavity, whose left vertical wall has been divided into upper and lower parts in the ratio of 4:6. The upper section is semitransparent window, while lower section is isothermal wall at a temperature of 296K. A collimated beam is irradiated with different values (0, 100, 500 and 1000 W=m2) on the semitrans- parent window at an angle of 450. The cavity is heated from the bottom by arXiv:2007.12484v1 [physics.flu-dyn] 23 Jul 2020 convective heating with free stream temperature of 305K and heat transfer coef- Email address: [email protected] (Pradeep Kumar) Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar ficient of 50 W=m2K while right wall is also isothermal at same temperature as of lower left wall and upper wall is adiabatic. All walls are opaque for radiation except semitransparent window. The results reveal that the dynamics of both the vortices inside the cavity change drastically with irradiation values and also with the boundary conditions on the semitransparent window. The temperature and Nusselt number increase multifold inside the cavity for combinedly conduc- tively and radiatively adiabatic condition than the only conductively adiabatic condition on the semitransparent window. Keywords: Semitransparent window; Natural convection; Collimated beam; Symmetrical cooling; Irradiation; Bottom Heating; 1. Introduction The knowledge of heat transfer in buoyancy driven flows is crucial in many natural and engineering applications, like HVAC systems, electronic cooling. In recent years, there has been increasing interest of the researchers in the analysis 5 of multi-physics problems involving all modes of heat transfer. Many researchers [1, 2, 3, 4, 5] investigated the buoyancy driven flows in 2D enclosures heating from bottom and cooled either one side or both sides. Acharya and Goldstein [6] studied the effect of internal energy sources on the natural convection in an inclined square box. Osman et al. [7] investigated the 10 effect of non-newtonian fluid (i.e Bingham fluid) on natural convection, it was observed that Nusselt number was smaller in Bingham fluids than Newtonian fluid for the same Rayleigh and Prandtl numbers. A comprehensive review on natural convection has been performed by Rahimi et al. [8] and Das et al. [9]. The interaction of radiation with natural convection has been investigated 15 by Mondal and Mishra [10], Liu et al. [11] and Kumar and Eswaran [12] in a 2- dimensional cavity with different optical thicknesses and reported that there was 2 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar an considerable effect of radiation on the natural convection. The square cavity has been selected in first two works whereas slanted cavity was used by Kumar and Eswaran. Lari et al. [13] also showed that the radiation has considerable 20 effects on the natural convection even at low operating temperatures. Further studies on the effect of different geometrical shapes, like square, triangle and circle at the centre of the square cavity on the combined radiation and natural convection were performed by Mezrhab et al. [14], Sun et al. [15] and Mukul et al. [16], whereas, the study with circle inside the circular cavity has been 25 performed by Xu et al. [17]. The discreate ordinate (DO) method [18] was most acceptable and popular method to solver radiative heat transfer equation before finite volume discreate ordinate method (fvDOM) [19, 20] and now the researchers mostly use fvDOM in the radiation field. The performance analysis of various methods for radiation 30 transfer equation (RTE) like FVM, DOM, P1, SP3 and P3 have studied by Sun et al. [21], and reported that the P1 method consumed minimal time than the other methods, however, it suffer in accuracy at low optical thickness of medium whereas FVM gave better accuracy compared to DOM but took more computation time. Assessment of natural convection with radiation for different 5 35 aspect ratios ranging from 1 to 6, and range of Rayleigh numbers 1:60 × 10 to 4:67 × 107 in rectangular enclosure have been studied numerically in [22] and heat transfer correlations has been formulated. It has been reported that the mean Nusselt number increased to 73.35 from 23.63 for aspect ratio 1 to 6. Webb and Viskanta [23] performed an experiment to study the natural con- 40 vection with water as working fluid in an rectangular enclosure irradiated with collimated beam from a side and kept opposite wall on constant temperature, whereas other walls were kept adiabatic. The results showed that there was formation of hydrodynamic boundary layers at the vertical walls. The authors 3 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar have also analysed numerically the same problem for the prediction of internal 45 heating in the fluid. Anand and Mishra [24] investigated the radiative transfer equation for vari- able refractive index for participating media. Ben and Dez [25] provided an exact expression for the radiative flux for the emitting and absorbing semi- transparent medium which has a linear refractive index variation. The above 50 researchers mostly considered the combined diffuse radiation with natural con- vection, however, little work is available on collimated radiation. Apparently to the present authors knowledge there is no much work focusing the interaction of collimated beam radiation with natural convection. Also, the walls of the geometries in all above works have been considered 55 as opaque which does not allow any energy enter into domain through radi- ation mode of heat transfer. Whereas, semitransparent wall allows energy to enter into the system by radiative mode of heat transfer but may or may not allow to leave the system. This arises two heat flux boundary conditions on the semitransparent window and such conditions are mostly encounter in practical 60 scenarios. In the present work two heat flux boundary conditions on the semi- transparent wall for natural convection problem including radiation have been investigated. This paper is outlined as follows; section 2 describes the prob- lem statement followed by mathematical modelling and numerical schemes in section 3. The validation and grid independent test are explained in section 4 65 and 5, respectively. Section 6 elaborates the results and discussion.Finally, the conclusions of the present numerical study are provided in section 7. 2. Problem statement Figure 1 depicts the geometry for present study. The ecludian co-ordinate axis are along horizontal and vertical walls of cavity and origin is at lower left 4 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar Figure 1: Schematic diagram of cavity for the present problem with collimated beam incident at an angle of 450 70 corner of the cavity. The gravity force acts in negative Y direction. The left wall of the cavity has been divided into two upper and lower parts in the ratio of 4:6, where the lower part of the cavity is opaque while upper part is semi- transparent (for the radiation energy) and the rest walls are considered to be opaque. The thermal conditions on the lower left and right vertical walls are 75 constant temperature at 296 K while upper wall is thermally adiabatic. The cavity is heated from bottom by convective heat transfer with heat transfer co- efficient of 50 W=m2K and free stream temperature is 305 K. The combined radiation and natural convection are considered inside the cavity. The working fluid of the cavity is considered to be transparent for radiation energy and flow 5 80 is governed buoyancy force corresponds to Rayleigh number 10 and Prandtl number (Pr=0.71). A collimated beam enters through upper left wall inside the cavity at an angle of 450. This semitransparent wall may or may not allow the radiation energy to leave the cavity assuming being low conductivity of semi- transparent material, the energy does not leave the system through conduction 85 mode of heat transfer. Based on these two scenarios two thermal conditions on the semitransparent wall are considered case (A). Conductively Adiabatic: The semitransparent wall allows to leave 5 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar energy through radiation mode but not by conduction mode, i.e qc = 0, case (B) Combinedly conductively and radiatively adiabatic wall: The semi- 90 transparent wall does not allow the energy leave neither by conduction nor by radiation heat transfer, i.e qc + qr = 0. The effect of above two boundary conditions on the fluid flow and heat transfer characteristics inside the cavity has been studied for irradiation values of 0, 100, 500 and 1000 W=m2. 95 3. Mathematical formulation and Numerical procedures 3.1. Mathematical formulation The following assumptions have been considered for the mathematical mod- elling of the problem; 1. Flow is steady, laminar, incompressible and two dimensional. 100 2. Flow is driven by buoyancy force that is modeled by Boussinesq approxi- mation.
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